Theory of damping in forward scattering photoelectron diffraction due to thermal spherical wave effects

JOURNAL OF ELECTRON SPECTROSCOPY and Related Phenomena

ELSEVIER

Journal of Electron Spectroscopy and Related Phenomena 88-91 (1998) 527-531

Theory of damping in forward scattering photoelectron diffraction due to thermal spherical wave effects Takashi Fujikawa a'*, Kentaro Nakayama a, Takumi Yanagawa b aFaculty of Science, Chiba University, Yayoi-cho, 1-33, lnage, Chiba 263, Japan bFaculty of Engineering, Yokohama National University, Yokohama 240, Japan

Abstract

Numerical calculations illustrate the importance of spherical wave effects in high energy ARXPS (Photoelectron Diffraction) spectra. In particular, spherical wave effects play an important role in the small-angle scatterings which predominate in ARXPS processes. In forward scattering both static and dynamical spherical wave effects play an important role, even in the high energy region. © 1998 Elsevier Science B.V. Keywords: ARXPS; Photoelectron diffraction; Spherical wave effects; Thermal effects

1. Introduction Plane wave approximation for the study of ARXPS (or Photoelectron Diffraction) spectra works well and has been widely used [1]. However, spherical wave correction cannot be neglected any more for the systems in which an X-ray absorbing atom is surrounded by heavy atoms, and several efficient methods have been developed to include those corrections in ARXPS theory [2,3]. So far the spherical wave effects on D e b y e - W a l l e r factors have been studied by Brouder et al. for EXAFS analyses based on oneparameter Lie group approach [4,5]; the relation to the plane wave Debye-Waller factor is not clear in their formula. A different approach has been developed to handle the Debye-Waller factors in EXAFS, including spherical wave correction by Rennert [6,7]. The temperature effects in photoemission spectra have rarely been studied in comparison with those in EXAFS. Only a few works are * Corresponding author.

found [1,8]. Recent work by Fritzsche et al. [9] uses the same formula as used by Brouder and Goulon [5] to study the effects of thermal motion of atoms in ARXPS spectra. So far we have developed a new approach based on Brouder's and Fritzsche's formula [4,5,9] to include both spherical wave corrections and the anharmonic vibration effects in ARXPS and EXAFS spectra, which are not limited to isotropic systems [10]. There we applied partial summation to get a plane wave (PW) part which does not disappear even for large kR limit, and a spherical wave (SW) part which disappears for that limit. Using this method, the thermal spherical wave effects on EXAFS were studied in detail [10]. In comparison with EXAFS spectra, small-angle scattering predominates in ARXPS spectra, so careful study is necessary on the thermal effects in ARXPS spectra. In this work some numerical examples are shown to illustrate the importance of the SW correction in ARXPS analyses [11]. Further extensive applications are given to discuss the importance of these dynamical spherical wave effects.

0368-2048/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PH S0368-2048(97)00156-4

528

T. Fujikawa et al./Journal of Electron Spectroscopy and Related Phenomena 88-91 (1998) 527-531

2. Basic theory The photoelectron current intensity l(k)¢ is given in terms of direct photoemission amplitude Zl, single scattering photoemission amplitude Z2, and double scattering term Z3 and so on. I(k)c ~ IZI +Z2 -{-Z3 -{-''12

(1)

Atomic contribution IZ112 has no effect from thermal motion. The lowest order scattering term Z~Z2 has thermal effects due to thermal atomic vibration. The z-axis propagator expansion is important to efficiently treat spherical wave effects in the high energy region [12]. For that purpose we apply the rotation that takes the vector R ° into z axis; a set of Euler angles of this rotation is abbreviated a s / ~ . The propagator is then written

GL"L,(kR °) =

~.D~"~(R=)D~,~(R.)*g~,,t, (R°) ,a

(2)

where M is defined in Refs. [5,9], and (X) stands for the thermal average of X defined by (X)=Tr[X exp(-f3n)]/Tr[exp(-13H)] where n is the vibration Hamiltonian of the system. As M x, M y, M z commute with each other, we can apply cumulant expansion W = e x p [ ~ (ik)n . . . . . ] ,= 1 n-----~. ~ [ u ' t i v l - 1~1)}')c

We rewrite M - I~1 as (1~° - !~)1 + ( M - R J^) 0 in order to separate PW D e b y e - W a l l e r factor from the SW D e b y e - W a l l e r factor as used in EXAFS study [10]; the SW factor comes from ( M - l ~ ° l ) as demonstrated later. We fully estimate the second order cumulant including spherical wave effects whereas we approximate the other cumulants by the plane wave term because the second order cumulant is the most important. Within this approximation we have WLL, = WpW(~LL, -- ~WLL, / 2 )

where Dlm~ is an element of rotation matrix: the dominant contribution comes from gO when kR ° >> 1, and the next order from g + i. From this discussion we have an approximate expression for Z~Z2 after we substitute Eq. (2)

Z~Z2 =exp{ikR°(1-cosO°)}/(R °) ~, ~, YL(k)*

(6)

where Wpw is the plane wave thermal factor, which is independent of L and L'. 6W is the dynamical spherical wave correction term due to thermal vibration which disappears in the limit kR ° ---, ~ (plane wave limit),

~W =k 2

mc LLI

(5)

Z i,j=x,y,z

(uiui){ 2(R~ ^o - k^ ) i ( M - R^oJ ) j

{fl'(oO)yL, ( R ° ) - f ~ ( O ° ) ( e i 4 J " D ~ , l ( e a )

+ (M - l~°l)i(M - R°l)j}

- e-iep"Dl~, _ 1(/~-)) +'"" }M/~LcML'L, exp ( - ik.u)JL,L, (u)( - 1)r +l~

(3)

The effective spherical wave scattering amplitudes f~ a n d J ~ at site a are defined in Ref. [11]. We can show that the amplitude f~(O~) is reduced to the ordinary scattering amplitude in the high energy limit, kR ° >> 1, and f-t~ vanishes . in that limit, which arises from helical waves during the propagation from atom A t o c~. To take the thermal effects into account, we should calculate the thermal average of the last two factors including u in Eq. (3) W = (exp {iku.(M - 1~1)})

(4)

(7)

At first sight it can be difficult to understand how 6W disappears in the high energy limit because 6W includes no information about kR °. If the amplitudes represented by f~ are independent of l, we obtain the relation

~. YL(R)(6W)LL, = 0 L

(8)

From the above discussion we can say that the spherical wave thermal factor arises from differences in the value of f~ which are finally due to the static spherical wave effects. From the above equations, we have the explicit expression for the lowest order term B including the dynamical plane and spherical wave effects, which can be written by the sum, (B)=(B)pw+(B)sw in

T. Fujikawa et al./Journal o f Electron Spectroscopy and Related Phenomena 8 8 - 9 1 (1998) 527-531

terms of Qtc LL' =

Xm,.M[L,.MvLc,*

(B)pw= c¢(:pA)X-~Re[exp{

First we study c(2 x 2) CO on Ni (001) surface. This adsorbate system has been extensively studied by Orders et al. [13]; CO molecules adsorbed on a-top site perpendicular to the (001) surface. Fig. 1 shows the energy dependence of the calculated XPS spectra (B) + (C) from C 1s core level for small take-off angle Ot = 13.0 ° in [100] direction, where C = [Z212. In this direction the emitted electrons are strongly influenced by the forward scattering from the oxygen atom of the nearby CO molecule. This figure shows the two spherical wave effects, the static and the dynamical SW effects: the former defined by the difference of the intensities in the static PW and in the static SW methods, the latter by the difference of the intensities in the static SW and the dynamical SW methods. As the C O distance is rather large, 5.11 .~, and the scatterer is not so heavy (oxygen), those spherical wave effects are not so pronounced. As observed in the normal photoemission from the same system [14], both of the spherical wave effects are much larger due to intra-molecular scatterings than those found in the small take-off angle detection due to large distance inter-molecular scatterings. Fig. 2 shows the same spectra as Fig. 1 except for normal emission for different temperatures, 100, 295 and 500 K, although all CO molecules should desorb at 500 K. To emphasize the thermal effects we consider this fictitious adsorbed molecule even at 500 K.

ikR° (1 -cosO°)} LL' ~" YL(k)*

0 ^0 ~1' 0 {f/~l' (Oo,)YL,(Ro~)+f o,(O,~)(ei4~,O mt' ' , l ( e c ~ )

-e-i%Dl'm', -, (/),~))+ "'" } Q~L', Wpw] ( B ) s w = - c,(*A) y"

(9)

-~ Re[exp{ikR°(1-c°sO°)}

X YL(k)*(6W)L,L,f~'(O°)YL,(II°)Q~L,Wpw]

529

(I0)

LUL I

Eq. (9) includes thermal effect through scalar factor Wpw, but Eq. (I0) includes both Wpw and the spherical thermal factor 6W which vanishes at static limit. The second term of wavy-parenthesis in Eq. (9) describes the lowest order helical spherical wave correction in the static limit, which vanishes in the forward scatterings [3].

3. Results and discussion In this work two adsorbed systems are considered.

Ni(O01 ) - c ( 2 x 2 ) - C 0 theta=l 3 degree ..... Static PW I phi:O degree ......... Static SW I 0.02 0.01 "~ I Static SW effect I - - SW(295K) "~

o.ms

-~

0.01

-k .......

................................. i.................... i........................... i - ~ :

. .r................ i.~.x

. i '....~. . . . . .

- i 'I

'

o.oo,

.......

'~ ' ~

................. ......

~,~ o.oos

,.~,~ ~...................... i . . . . . . . ~

I

V

0

m

m

. . .........

',....:

,'

o

+

_o.o,

A

I~ -0.005

-0.015

Dynamical SW -0.01

effect| '

400

600

800 1000 energy(ev)

1200

-0.02

1400

Fig. 1. The energy dependence of the photoemission intensity (B} + (C) from C 1s level of adsorbed CO on Ni (001 ) surface at 295 K. Take-off angle is 13.0 ° in the [100] direction.

530

T. Fujikawa et al./Journal of Electron Spectroscopy and Related Phenomena 88-91 (1998) 527-531

Ni(001)-c(Zx2)-CO Normal emission r,

ID l f

.......

f ' II

,,

o.ool A (.9 V + A

I

¢~

l I l#,.L....i

0.002

J ..... SW(OK) J ..... SW(10OK) J ......... S W ( 2 9 5 K ) SW(SOOK)

[ ~

'i ..~..

J .I

~

~

~ J

-

-

I

I b

~ • ~ j

I

e ~

f

I

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,"

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I

I

f

', , ,!............. , , , ,~.' . t,.~..~, , ;................ , . - . ,),...'....~:..........................,.%' , ...................... ,,, j~ ,~r:~. ,.....~..Li..',.

i"",i;."~.~i

I

%1

't~i.

..."....... '!-"

i

i~

i

I~

-~

.

~

•

'.,i,.'

•

', ,.' .

0

;\",a.'f

i

: il

"

",, ~ ," ,,

,..

V -0.001 400

600

800

1000

1200

energy(ev)

1400

Fig. 2. A s i n Fig. 1 except ~ r n o r m a l e m i s s i o n .

Under this forward scattering condition, the plane wave thermal factor cannot contribute to reduce the photoelectron diffraction oscillation in the isotropic approximation, where we have Wpw=exp[-k2(u2)(1-cosOc~)]. Contrary to this observation, the dynamical SW effects give rise to the remarkable temperature dependence even in the forward direction. In comparison with the result in Fig. 1 the C - O distance is 1.15,~ which is much shorter than that considered in Fig. 2, 5.11 ,~. Both static and dynamical SW effects are expected to be larger than those found in Fig. 1. The photoelectron diffraction spectra of molecular N 2 o n Ni (100) is also interesting [15]. Owing to the perpendicular adsorption geometry, the two nitrogen atoms are inequivalent and two chemically shifted N I s peaks are observed. The component with the lower binding energy corresponds to the outermost nitrogen atom. First we study the PW thermal effects. Fig. 3 shows the polar angle dependence of the photoemission intensity from the inner nitrogen atom for temperatures of 0, 200, 298 and 400 K. At normal emission (0 = 90°), as expected, all photoemission intensities are the same. Even if the scattering angle is large enough up to 50 ° (0 = 40°), the temperature effect is small. Next we study the SW thermal effects for this system. Fig. 4 shows the same ones as those shown in Fig. 3 except for the SW calculations. In comparison with the results in Fig. 3, remarkable temperature dependence is found even in the forward scatterings (0 = 90°). The experimental result has

not been compared with these calculated ones because neither the details of the X-ray polarization direction nor the detection mode such as sample rotation are mentioned in Ref. [15]. However, overall polar angle dependence in Fig. 4 is quite similar to the observed one.

intensity of the inner nitrogen atom (PW) 0.03

~N~i

1.....StaticPW

~'_~O.02S - ~ i

c=

............... J. . . . . PW(2OOK)

~

"~ o.o2

I .........PW(298K)

)

0.005

i 90

80

70

60

50

40

polar angle Fig. 3. Polar angle dependence of the N ls level on the inner nitrogen (directly adsorbed on Ni surface) at different temperatures. The P W thermal effects are considered.

T. Fujikawa et aL/Journal of Electron Spectroscopy and Related Phenomena 88-91 (1998) 527-531

Acknowledgements

intensity of the inner

nitrogen 0.025.

atom ~

"%

(SW) --q

'

w/

k. I ..... Static S ',~ I ..... SW(ZOOK) I o.o2 ............:'~.................I ......... SW(Z98K) I -~ i', I--SW(4OOK) ~ ......:~ ~.

J !

i i

i~.,

i

%, i

"~ O)

i

t--

i

• --

0.01

- ....................

I

?l

ik

0.005 90

80

References I,L, •

11,

L......~

~.~r.

~...l

i t"

One of the authors (T.F.) is grateful for financial support from the RICOH Research and Development Center.

!

. • .....~, •

[ ................

531

i

................ i ..............

;

i

i

i

i

i

70

60

50

__ 40

polar angle Fig. 4. Polar angle dependence of the N Is level on the inner nitrogen at different temperatures. The SW thermal effects are considered.

4. Conclusion The dynamical SW effects give rise to a remarkable reduction of the XPD intensities even in the forward scatterings, which is in contrast to the PW thermal effects. To closely compare the observed results, we should average the calculated results over the finite acceptance angle. The strong dynamical SW effects would be smeared out to some extent.

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